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ARITHMETICK is the science of Numbers ;* and comprises the following principal rules, viz. I. Notation, or NUMERATION; II. ADDITION ; III. SUBTRACTION ; IV. MULTIPLICATION ; and V. Division. The four last are called simple, when the numbers are all of one denomination; compound, when the numbers are of different denominations.
These five rules are called principal or fundamental, because the whole art of arithmetick is comprehended in their various operations.
1. NUMERATION teaches us to read or write any sum or num. ber, by means of the following ten characters, called figures of Cipher. One. Two. Three. Four. Five. Six. Seven. Eight. Nine, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
2. The value of a figure, when alone, is called its simple value, and is invariable. Figures have also a local value, which varies
* Number is either a unit, or a collection of units. A single thing is a unit, or one. One and one are two. One and two are three. And thus, by constant addition, all numbers are generated.
† These figures, which are of Arabic or Indian origin, were introduced into Europe by the Moors, about the year 1150; they were formerly ali called ciphers, whence it came to pass that the art of arithmetick was called ciphering:
1. What is Arithmetick?- -2. What are called its fundamental rules ?3. What does numeration teach ?----4. How many figures are used to represent numbers ?5. How do you determine their value?
according to the place in which they stand. In a combination of figures, reckoning from the right hand to the left, the figure at the right, or in the first place, represents its simple value ; that in the second place, ten times the simple value ; that in the third place, ten times the value of that in the second place ; and so on, in a tenfoid increase.
EXAMPLE. Write down the sum 4 4 4 4. The first figure at the rigbt, in the place of units, has its simple value, or the same as if standing alone-four. The second, in the place of tens, signifies four tens, or forty. The third figure, in the place of hundreds, signifies four hundred, or ten times its value in the place of teps. The fourth figure is in the place of thousands, bearing ten times its value in the place immediately preceding.
3. A cipher 0 though of no signification itself, when placed on the right hand of figures, in whole numbers, increases their value in the same tenfold proportion. Thus 9, signifies only nine, its simple value. Place a cipher on the right, (90) it becomes ninety ; and by placing two ciphers at the right, thus, (900) it becomes nine hundred.
9 nine units, or nine. 9 8 nine tens, and eight units, or ninety-eight. 98.7 nine hundreds, eight tens, and seven units, or 8 7 6 nine thousand, 876.
(nine hund. 87. 7 6 5 ninety-eight thousand, 765. 6 5 4 nine hundred eighty-seven thous. 654. 5 4 3 nine millions, 876 thous. 543. 4 3 2 ninety-eight millions, 765 thous. 432. 3 2 1 nine hundred 87 millions, 654 thous. 321.
9 9. 8 9 8 many 8 7 6 ng 6 5 6 5 4
Note.-Six places of figures, beginning on the right, are called a period; but they are commonly divided into half periods of three figures each. This division enables us to read any number of figures as easily as we can read the first period.
6. In a combination of figures, how do you ascertain their value ?7. What is the nature of the cipher ? -_-8. What is the period of Numeration ?
RULE. —Commit the words at the head of the Table, viz. units, tens, hundreds, &c. to memory; then, to the simple value of each figure, join the name of its place, beginning at the left hand and reading towards the right. More particularly-1. Place a dot under the right hand figure of the 2d, 4th, 6th, 8th, &c. half periods, and the figure over such dot will, universally, have the name of thousands. 2. Place the figures 1, 2, 3, 4, &c. as indices, over the 2d, 3d, 4th, &c. period : These indices will then show the number of times the millions are involved--the figure under 1 bearing the name of milliðns, that under 2, the name of billions, or millions of millions ;that under 3, trillions, or millions of millions of millions.
1 913,208.000,341.620,057. 219,356, 819,379. 120,406. 129,763
Note.—Billions is substituted for millions of millions ; Trillions, for millions of millions of millions ; Quatrillions, for millions of millions of millions of millions. Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, UndeciHions, Duodecillions, &c. answer to millions so often involved as their indices respectively denote.
The right hand figure of each half period has the place of units, of that half period; the middle one, that of tens, and the left hand one that of hundreds.
APPLICATION.-Let the scholar now read, or write down in words at length, the following numbers :
8 437 709.040 3.476.194 17.184.397.647 17 3.010 879.096 84.094.007 184.108.40.206 129 76.506 4.091.875 690.748.591 500.098,422.700
Write down, in proper figures, the following numbers :-
9. How are numbers commonly divided ?--10. Of what use is this division ?-11, What is the rule for numeration ?
ROMAN NOTATION is the method of representing numbers by Letters ; and is now chiefly used to number the Charters of books, &c. Seven letters are used for this purpose, viz. I, V, X, L, C, D, and M.--I, signifies 1 ; V, 5; X, 10; L, 50 ; C, 100 ; D, 500; and M, 1000.
One. XV, Fifteen. CC,
Two Hundred. III, Two. XVI, Sixteen. CCC,
Three Hundred. III. Three. XVII, Seventeen. ICCCC, Four Hundred. IV. Four. XVIII, Eighteen. D, or 15, Five Hundred. v. Five. XIX, Nineteen. DC,
Six Hundred. VÍ, Sis.
xx, Twenty. DCC, Seven Hundred. VIÍ, Seven. XXX, Thirty DCCC, Eight Hundred VIII, Eigbt. XL, Forty. DCCCC, Nine Hupdred. IX, Nine. L, Fifty. M, or Clo, One Thousand, X, Ten. LX, Sixty. 13,
Five Thousand. XI, Eleven. LXX, Seventy. Iാാാ, ,
Fifty Thousand.* XII, Twelve LXXX, Eighty. 131023,
Five Hund Thous. XIIÍ, Thirteen. XC, Ninety. MDCCCXXVII,One Thous. Eight XIV, Fourteen.c, Hundred. Hundred and Twenty-seven.
* Sometimes thousands are represented by drawing a line over the top of the numeral letter : thus V represents five thousand, I fisty thousaud, CC two hundred thousand.
12. What is Roman Notation ?--13. What is its use ?-_-14. How many letters are used for this purpose? -----15. What number does each represent ?
RULES. 1. The number of a letter is doubled as often as it is repealed; thus, I, represents one ; II, two ; X, ten ; XX, twenty ; XXX, thirty.
2. A less literal number placed after a greater, augments the value of the greater ; if put before, it diminishes it. Thus, VI, is 6; IV, is 4 ; XI, is 11 ; IX, is 9, &c.
ADDITION. Addition is the putting together of two or more numbers, or sums, to make them one total or whole sum.
SIMPLE ADDITION Is the adding of several numbers together, which are all of one sort, or kind; as, 7 pounds, 12 pounds, and 20 pounds, being added together, make a sum total, or aggregate, of 39 pounds.
FULE. Place units under units, tens under tens, &c. : draw a line underneath, and begin with the units : After adding up every figure in that column, consider how many tens are contained in their sum, and placing the excess under the units, carrying so. many as you have tens to the next column of tens : Proceed in the same manner through every column or row, and set down the whole amount of the last row.
PROOF. Begin at the top of each column, and add the figures downwards, in the same manner as they were added upwards, and, if it be right, this aggregate will be equal to the first amount. Or, cut off the upper line of figures, and find the amount of the rest ; then if this amount and upper line, when added together, be equal to the sum total, the work is supposed to be right.
* This rule is founded on the known axiom, that “the whole is equal to the sum of all its parts." The method of placing the numbers, and carrying for tens, is evident from the sature of notation ; for any other disposition of the numbers would alter their value; and carrying 1 for every 10, from an inferior to a superior denomipation, is evidently right ; because one unit in the latter case is equal to the value of ten units in the former,
16. How are other numbers represented ?- -17. What is Addition ?- -18. What is Simple Addition ?-19. Repeat the rule.-20. Why do you carry for ten, in adding simple numbers ? -21. What is your method of proof?